Abstract In this paper, motivated by recent works on the study of the equations which model electrostatic MEMS devices, we study the quasilinear elliptic equation (Pλ) { - ( r α ⁢ | u ′ | β ⁢ u ′ ) ′ = λ ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - u ) 2 , r ∈ ( 0 , 1 ) , 0 ≤ u ⁢ ( r ) < 1 , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 . ${}\begin{cases}-(r^{\alpha}|u^{\prime}|^{\beta}u^{\prime})^{\prime}=\dfrac{% \lambda r^{\gamma}f(r)}{(1-u)^{2}},&r\in(0,1),\\ 0\leq u(r)<1,&r\in(0,1),\\ u^{\prime}(0)=u(1)=0.&\end{cases}$ According to the choice of the parameters α, β, and γ, the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the p-Laplacian, and the k-Hessian. We prove the existence of an extremal parameter λ ∗ > 0 ${\lambda^{\ast}>0}$ such that, for λ ∈ ( 0 , λ ∗ ) ${\lambda\in(0,\lambda^{\ast})}$ , there exists a minimal solution u ¯ λ ${\underline{u}_{\lambda}}$ and, for λ > λ ∗ ${\lambda>\lambda^{\ast}}$ , there exists no solution of any kind. We also study the behavior of the minimal branch of solutions and we prove uniqueness of solutions of ( P λ ∗ $P_{\lambda^{\ast}}$ ) for β > - 1 ${\beta>-1}$ .